Thermodynamic basis for the demarcation of Arctic and alpine treelines

At the edge of alpine and Arctic ecosystems all over the world, a transition zone exists beyond which it is either infeasible or unfavorable for trees to exist, colloquially identified as the treeline. We explore the possibility of a thermodynamic basis behind this demarcation in vegetation by considering ecosystems as open systems driven by thermodynamic advantage—defined by vegetation’s ability to dissipate heat from the earth’s surface to the air above the canopy. To deduce whether forests would be more thermodynamically advantageous than existing ecosystems beyond treelines, we construct and examine counterfactual scenarios in which trees exist beyond a treeline instead of the existing alpine meadow or Arctic tundra. Meteorological data from the Italian Alps, United States Rocky Mountains, and Western Canadian Taiga-Tundra are used as forcing for model computation of ecosystem work and temperature gradients at sites on both sides of each treeline with and without trees. Model results indicate that the alpine sites do not support trees beyond the treeline, as their presence would result in excessive CO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2 loss and extended periods of snowpack due to temperature inversions (i.e., positive temperature gradient from the earth surface to the atmosphere). Further, both Arctic and alpine sites exhibit negative work resulting in positive feedback between vegetation heat dissipation and temperature gradient, thereby extending the duration of temperature inversions. These conditions demonstrate thermodynamic infeasibility associated with the counterfactual scenario of trees existing beyond a treeline. Thus, we conclude that, in addition to resource constraints, a treeline is an outcome of an ecosystem’s ability to self-organize towards the most advantageous vegetation structure facilitated by thermodynamic feasibility.

MLCan has been validated for numerous sites across the Western Hemisphere 21,[35][36][37][38][39][40][41] . To apply MLCan to the harsh winter conditions of Arctic and alpine ecosystems, we included new parameterizations for peat soils and switches to start and stop photosynthesis to simulate dormancy during winter. Cold temperatures, freezing soils, the hibernation behavior of vegetation to not perform photosynthesis in the winter, and the varying behavior of soils with permafrost conditions required updates to model formulation in order to apply MLCan to this new region. The updates to MLCan are validated for all sites in Figures S1 -S3.
Soil. Soil properties, such as sand, clay, and organic material content and hydraulic and thermal conductivity are parameterized and held constant throughout the model simulations. Due to the presence of peat soils in the Arctic, we implemented formulations for the thermal conductivity of peat soils instead of basing them entirely off of sand/clay percentages. The thermal conductivity model was based on equations from Zhao et al 42 , and parameterizations for thermal and hydraulic conductivity were based on Wu et al 43 and Krogh et al 27 . To further take into account the behavior of Arctic permafrost, the model was altered to turn off plant-soil uptake when a given soil layer was frozen, specifically when the soil temperature within the layer of the subsurface was determined to be below -1 • C.
Canopy. We created dynamic switches to stop photosynthetic activity during winter when the mean air temperature in the canopy over the previous 24 hours drops below a certain threshold (-7 • C) and restart when it returns above a certain threshold (3 • C) 44 . Based on site-specific literature, different photosynthesis stop (-3 • C) and start (5 • C) thresholds were used for NR1 vegetation 11 . Additional constraints preventing photosynthesis from occurring when the top layer of the soil is frozen or the snow depth is greater than the canopy height are also included 45 . We do not close the stomata or change the respiration routine since literature indicates that respiration can occur during winter, even when photosynthesis is not occurring 46 . The periods of time when photosynthesis was active were validated based on site-specific literature when available.
Further, due to the extended periods of snowpack in the regions studied, new parameters were created to demonstrate the change in canopy reflectance with snow 47 . These new snow reflection coefficients vary based on the surface (i.e., canopy, peat soil, sandy soil) and the type of radiation (i.e., PAR, NIR) and are summarized in Table S1.
Ecosystem-wide. Due to the sensitivity of latent heat of vaporization (Lv) in colder regions, we implemented dynamic Lv based on air temperature 48 rather than keeping it as a static parameter. We implemented a bi-directional formulation for estimating temperature and vapor pressure values from observed fluxtower measurements. Since counterfactuals were constructed at the alpine/Arctic sites, fluxtower measurements above alpine/Arctic shrubs were located below the height of the simulated trees and, consequently, the ecosystem height (see Table S1). This ecosystem height was used as the upper bound of the control volume for each site pair such that shorter canopy and flux tower heights (i.e., alpine/Arctic tundra or meadow) could be compared directly with taller ecosystems (i.e., subalpine/subArctic forest). Since fluxtower measurements were collected below the ecosystem height for the alpine/Arctic sites and above the ecosystem height for the subalpine/sub-Arctic sites, the new formulation calculates the estimated temperature and vapor pressure deficit at the ecosystem height from the observed values from either above or below using similarity theory.

Supplementary Figures
The main text presents several figures with data or results from only one or two of the locations studied. This includes the average daily work by scenario throughout the year; the daily snow depth, average daily photosynthetic rate, and average daily respiration rate for each scenario throughout the year; and 3-D plots of temperature gradient, leaf area index, and work for all scenarios. The data for the remaining sites are presented in Figs. S5 -S7, respectively.   Figure 6 of the main text for information on the Western Canadian Taiga-Tundra scenarios. The negative of the resultant temperature gradient is plotted. Thus, positive values refer to negative temperature gradients such that larger values indicate stronger declines in temperature from the earth surface to the atmosphere. Negative values indicate positive temperature gradients, or temperature inversions. The 3D views show the transition from flatter curves to greater marginal increases in work with increases in temperature gradient as more LAI is modeled for each set of environmental conditions (i.e., alpine, subalpine). The simulated alpine forest scenario exhibits considerable negative work values since the LAI is beyond the supported limit of the local environmental conditions.